+713 Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$
+88 Let's first solve for ab:
\( (a + b)^2 = a^2 + b^2 + 2ab\)
\(1^2 = 2 + 2ab\)
\( -1 = 2ab\)
\(ab = -\frac{1}{2}\)
Now, we can solve for a^3 + b^3:
\((a^2 + b^2)(a + b) = a^3 + b^3 + a^2b + b^2a\)
\((2)(1) = a^3 + b^3 + ab(a) + ab(b)\)
\(2 = a^3 + b^3 + ab(a + b)\)
\(2 = a^3 + b^3 + -\frac{1}{2}(1)\)
\(a^3 + b^3 = 2 + \frac{1}{2}\)
\(\mathbf{a^3 + b^3 = 5/2}\)
+88 Let's first solve for ab:
\( (a + b)^2 = a^2 + b^2 + 2ab\)
\(1^2 = 2 + 2ab\)
\( -1 = 2ab\)
\(ab = -\frac{1}{2}\)
Now, we can solve for a^3 + b^3:
\((a^2 + b^2)(a + b) = a^3 + b^3 + a^2b + b^2a\)
\((2)(1) = a^3 + b^3 + ab(a) + ab(b)\)
\(2 = a^3 + b^3 + ab(a + b)\)
\(2 = a^3 + b^3 + -\frac{1}{2}(1)\)
\(a^3 + b^3 = 2 + \frac{1}{2}\)
\(\mathbf{a^3 + b^3 = 5/2}\)
